Graph energies of connected graphs

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Numbers

$G$

$\mathcal E(G)$

A_:

comment: This is the complete graph $K_{2}$.

BW:

2.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145

comment: This is the complete bipartite graph $K_{1,2}$.

Bw:

comment: This is the complete graph $K_{3}$.

CF:

3.464101615137754587054892683011744733885610507620761256111613958903866033817600074162292373514497151

comment: This is the complete bipartite graph $K_{1,3}$.

CL:

4.472135954999579392818347337462552470881236719223051448541794490821041851275609798828828816757564550

comment: This is the path $P_{4}$.

CN:

4.962388608184031245267074482433714361105490433996953456791786281627673844773530106026582365095707850

C]:

comment: This is the complete bipartite graph $K_{2,2}$.

C^:

5.123105625617660549821409855974077025147199225373620434398633573094954346337621593587863650810684297

C~:

comment: This is the complete graph $K_{4}$.

D?{:

comment: This is the complete bipartite graph $K_{1,4}$.

D@s:

5.226251859505506111426572693708748614335044753397078110195593467632324165844768203807414880770512573

D@{:

5.627213005296661630882455124721360484368562409530072110653088354444577673408278880813888259313606972

DBg:

5.464101615137754587054892683011744733885610507620761256111613958903866033817600074162292373514497151

comment: This is the path $P_{5}$.

DBk:

5.841619252963778989528394936201772181691914933456771936981350301637687873930815344619033490397272993

DB{:

6.040895836088439143157393944332025234449332756993754506055066285763845857942823557660177054423363062

DFw:

4.898979485566356196394568149411782783931894961313340256865385134501920754914630053079718866209280470

comment: This is the complete bipartite graph $K_{2,3}$.

DF{:

DIk:

5.595865303863626808126162157727995058694090255464955519723745292601494368151921196459313202005787212

DJk:

6.729857197628430364684743172119851451390231216405962831133190349826779811483147325378582591321301198

DJ{:

7.028273858670582145387549733924434956104952601490809184443342788418683314583547181516140817592433007

DK[:

6.428639486755070374830995401697160977815839274438998868662764633018256092866532549591942500683928704

DK{:

7.123105625617660549821409855974077025147199225373620434398633573094954346337621593587863650810684297

DLo:

6.472135954999579392818347337462552470881236719223051448541794490821041851275609798828828816757564550

comment: This is the cycle $C_{5}$.

DL{:

7.106932641439849271620054204194994275308998787610094381916855823365030930174096340478767938990008305

DNw:

6.354819361798569134901586241832665906713962235605703914510435759821252835127490673103397817754782655

DN{:

7.362661287209954750951324775777573098056549906376278871133809559524887491648443547650162111134346972

D]{:

6.472135954999579392818347337462552470881236719223051448541794490821041851275609798828828816757564550

D^{:

7.291502622129181181003231507278520851420518366164900360736668918402137646460567255520785772949087221

Dbk:

6.340172973252067445406511528850741850840279259646619907337454812773690100341494779351308368367151965

D~{:

comment: This is the complete graph $K_{5}$.

E?Bw:

4.472135954999579392818347337462552470881236719223051448541794490821041851275609798828828816757564550

comment: This is the complete bipartite graph $K_{1,5}$.

E?Fg:

5.818625822352818929219598264041055142953972376759860602673656935068888676893280775375622774447420654

E?Fw:

6.172260395302988182499243274477199753959975655153459902763466262220457154157809499885192523983773815

E?NG:

E?NW:

6.485204316658384165369415260377968108304001967574661845531330742101138747909920767103821641547539684

E?Nw:

6.685846165554340416109718021962283266803071289977970214792795219557662739625068406045771876985518948

E?]o:

6.472135954999579392818347337462552470881236719223051448541794490821041851275609798828828816757564550

E?]w:

6.818643672455629702997220862459284724425074936379497626878139369317389351185575786018775967872227308

E?^o:

6.469652731063006471994176723960381140254735347486275713849986057910157391816831967501130651573875401

E?^w:

6.892602538996459915510055544713000379505516624355759069718937885622573260677179338179838820195797976

E?~o:

5.656854249492380195206754896838792314278687501507792292706718951962929913848428155401550137310566291

comment: This is the complete bipartite graph $K_{2,4}$.

E?~w:

6.744562646538028659850611468218929318220264457982792367699877470565900721457404627027125365596788122

E@FG:

6.155367074350506805140581152073819648013404287075584854078312500749726576990181836909187443334210871

E@JW:

6.324555320336758663997787088865437067439110278650433653715009705585188877278476442688496216758600590

E@NW:

7.464101615137754587054892683011744733885610507620761256111613958903866033817600074162292373514497151

E@Nw:

7.711545013271977351697399500659702416275690538745588800623814926427537724208005134409573773047318314

E@QW:

6.898979485566356196394568149411782783931894961313340256865385134501920754914630053079718866209280470

E@Rw:

7.806423851823106574970432448114188467550628100088665319208433164164272348434497222916668503955372819

E@UW:

7.300563079745769490421724785881948628020580469976947594895153966802506808199823876529603885412847695

E@Vw:

7.980630624037818356259785136950205553660882817594318091970774715976421647095209526441539773975570397

E@YO:

6.987918414869868244200039072033918485058197847171216842924395512774829219651898276607540058612682973

comment: This is the path $P_{6}$.

E@YW:

7.207750943219353009888818556059560409327353297054857200428499385595459363423519724010891637748099323

E@]o:

7.806423851823106574970432448114188467550628100088665319208433164164272348434497222916668503955372819

E@]w:

8.018832738761591386610631737963339899784494878690034205398589556496812742468149398652463981267924772

E@^W:

8.118304317665228386691710807538655528101379879940575772334773933102136334644668256594244430217791047

E@^o:

7.937221616936039444767967084929373030558664019256204606045984859020947670580602617669582651811894493

E@^w:

8.289211745690182279716713083384735694668911097633051140776027498347070226897139745447987010093198975

E@`w:

7.342490027239563359100802631051322164323697912576091254810242135124860587773647822194172635512741266

E@ow:

7.191688434954841334000599958088537435893160330236880994896402687062374139651169785852916278911606541

E@vw:

8.106541021659371939086743316418276507203987495204182847276834605987974869953367425995714730899784461

E@~o:

7.252396137054587226752999100157248773655723120526134908903242911301486037695092019788484787918732210

E@~w:

8.282672231310728269551879833168888276688194050170545981623850279558514012078022759328958604765507125

EAMw:

7.665495618935063797756322036287182121396164814876813815045212087700168706753341842320606804855516022

EA]o:

7.766202102101374486312813698521798407680704392514778399975193977189272808177114659749583307852257383

EBYW:

6.633249580710799698229865473341373367854177091178707194117364292232969285218087693417686798256581302

EBYw:

7.217425947927284760186394673654378480572877889801090344492974153105434015104977780607804884149202046

EBZw:

7.362661287209954750951324775777573098056549906376278871133809559524887491648443547650162111134346972

EB]w:

8.252738179839737612464051373207287157554623241284084170960354821914988661088281495799106696274487338

EB^w:

8.302408025444418287473224762898412568297311819369112714278824318014405427302692234099350984896002587

EBhw:

8.045782680109072186491267601562466519328767905059070149638106215470524692600965253960111357678792824

EBjW:

8.685846165554340416109718021962283266803071289977970214792795219557662739625068406045771876985518948

EBjw:

8.670264050430885468402107495056076867304826990281515189985593048870382652028166563057944032014422476

EBnW:

8.944271909999158785636694674925104941762473438446102897083588981642083702551219597657657633515129100

EBn_:

8.138805226879111362292207869575203895224268732804641555922402585705256105953877243733115221389054374

EBnw:

8.739132120594849621507374998490704448889330471861664128739367324438100982256889153116876289267583598

EB~o:

7.620386170314852043228323625548326198059017885243619916429023858032260593433478290346689734936195818

EB~w:

8.584803170449242064607406715328122023224441024878576102185271672630531757157080873495281709748105697

ECXw:

7.523338239663346370427019854163306575741281777600115302451214085271832036701569405823716322919619631

EC\w:

7.738308315348773670383805702997256015893208880872407551781541807860154608643026642461734201597656333

EENg:

7.299261078610975623914039113910742892221665931773083591908072385707430757083736374533830672717314220

EFz_:

comment: This is the complete bipartite graph $K_{3,3}$.

EFzw:

7.532870967705409570985027514093138426965233756567403282873786952836213690429009296991866816226311875

EF~w:

8.324555320336758663997787088865437067439110278650433653715009705585188877278476442688496216758600590

EGNW:

7.664074803029532747057158884653540033462970954922327949207832494891403889989321265058799135569028912

EGcw:

7.415980409830338533606879652048765431723125679222008151999025725997866846644997402768184860464336890

EGdo:

7.549245799008978526396499275838955109331020673056416374699026785931968209056785323729425641799010412

EGdw:

7.996110202208088649296548044183282356109804636639527218028460526113526505690393759877014291267753958

EHQW:

7.465883060453301293440401790148364171688130886385878141588871469089663331556219585299366100151376852

EHUW:

7.123105625617660549821409855974077025147199225373620434398633573094954346337621593587863650810684297

EHuw:

8.354510433691246187788401585359558778391570287434694899176055214776519327053007578279328703803712386

EI]w:

8.058990926761212908838115834846649501974798530504108716962454683345877715253104825203913712418283488

EI_w:

6.602720495543138223533364444482982175157738827615102828121493293368231195114831269210201956142121332

EIe_:

comment: This is the cycle $C_{6}$.

EImo:

6.928203230275509174109785366023489467771221015241522512223227917807732067635200148324584747028994303

EImw:

8.046818857603472256363449670096762058722726648012348385626430368601773044836061219778305751385064794

EIno:

7.588281363810870083867019888914995428791358802425634930417824258987032738163709843802822516519548946

EInw:

8.367008185346933250462298986186914545934939727716523965011213296334176546664038375918551229919433391

EJYW:

7.790213031855061353869725602536475764102620319110589315638878498097052329235981808662338895609605228

EJ]w:

8.887495848379203243067883913332089078513561870548240015203006509128877100982839912866213975640921379

EJ^w:

9.068140393445463126483212946635742070269240186529834309262143445342071325533196542867782417849343863

EJeg:

8.292528739883944684658270131431140891024954258374657402464973434885330990741814151863067442169780297

EJmw:

9.123105625617660549821409855974077025147199225373620434398633573094954346337621593587863650810684297

EJnW:

8.761374662616505798953755051538595622962943212525710753837704533664513123196992781931969406533695082

EJnw:

9.276221012466257272166994242136815173338707924309142017682447983631636716250095373369971638541906003

EJqw:

8.704443792558203844348564992206169557842158505943969604542220567359967330513681795557752191807169533

EJ~o:

8.558904631537212739523092317723424403178210058006062496244089250360348058732807645131710872469640256

EJ~w:

9.493136493991434602315200217327588875522945350062650619343390089511901699328955007644280581127120015

EKNG:

7.656854249492380195206754896838792314278687501507792292706718951962929913848428155401550137310566291

EK\w:

8.707711570861656477116674698043461334364756173301840070083373106868484215602215603448230936081400335

EK]w:

8.857845879539719904560491040046454396527738352950486278443571922436654516035091859988808189129889247

EK^w:

9.234333538695950085007351467523637631750705453965059451783156359499421538367025967587646511778901190

EK`w:

7.535762184188808718471296330939752238636840905964136670315087039861977668057514493128118235045419170

EK~o:

8.744562646538028659850611468218929318220264457982792367699877470565900721457404627027125365596788122

EK~w:

9.656854249492380195206754896838792314278687501507792292706718951962929913848428155401550137310566291

ELpw:

8.957679683723196469180846030462219229876300364763588355076722450660040818668371691885174680674497794

ELrw:

9.371115440565935589212915486874335254813131680536391705407179625322962606190239851908547682966845020

ELv_:

EL~o:

8.663237301241977881970097591651014317600787295629976118492720710385104237287242176217299779758002187

EL~w:

9.474121328551627543198229735454653111600656048068112031877802560055097191813384324490153379272557755

ENzw:

8.858814602077679498235114893473357941217628001603362900070613546266982993360655737824941126813517542

EN~w:

9.607084991365190977216903789249965914517164530450806808595762597256895230005584465079200660115220155

EPTW:

7.732396525018045010947988461973182812471801511545760923956833409722234152217432395652790061322564501

ER^W:

8.756632523518574745450033524416842167824236448466767125367357678789859778593661785616210345326052979

E]~o:

E]~w:

8.744562646538028659850611468218929318220264457982792367699877470565900721457404627027125365596788122

E^~w:

9.403124237432848686488217674621813264520420132621018885529272626668182758196876074289354302249869963

E_Nw:

8.213373930418942300937140671668059228136271707449090014846760917362711189270989274955147776282623690

E_]o:

7.564563854420212122516857986342222147547235306566398575893406308353486152439460940160291590543418702

E_]w:

7.418550718873845678458263334247781867040053478730873310775911706109062107227028689458838668987775115

E`Lw:

8.426782210190877937441515962573110255546985758387040718781980295871188396833727559392869250275773756

E`NG:

8.292528739883944684658270131431140891024954258374657402464973434885330990741814151863067442169780297

E`NW:

8.698355190948984576147920653150570895697895170588746921672812701273068274726596613790680694577927923

E`Nw:

9.203358263766308504928740720375662412400114174479743900804985469463782944844475863965579788572641874

E`]o:

8.818643672455629702997220862459284724425074936379497626878139369317389351185575786018775967872227308

E`]w:

8.472135954999579392818347337462552470881236719223051448541794490821041851275609798828828816757564550

EaMw:

8.182694489906029904975478778099991313301995815079856779410620038593184843719248132354145133606157821

Ebnw:

8.978577143620157525585873060909698482336131678965986231073132910341349868285232838686803378771021181

Ejmw:

8.828427124746190097603377448419396157139343750753896146353359475981464956924214077700775068655283145

Ek]w:

8.755422416913548648976315770728372961290390146882261791084206989648956280090617007302950962310014388

EoLW:

8.389762986525928263657805858015282898065057334380609371575904559128536166110804118691284157147071123

Es\w:

7.435482234928008190257458925909355675319276537159085057594381136795997924564337448889409938510939521

E~~w:

comment: This is the complete graph $K_{6}$.

Definition

For a finite simple connected undirected unweighted graph $G$ on $n\geq2$ vertices, the graph energy is $\mathcal E(G)=\sum_{i=1}^{n}|\lambda_i|$, where $\lambda_i$ are its adjacency eigenvalues, counted with algebraic multiplicity [1].

Parameters

$G$

— graph (a finite simple connected undirected unweighted graph with at least two vertices, identified by its Sage canonical graph6 string)

Formulas

(1)

$\mathcal E(G)=2\sum_{\lambda_i>0}\lambda_i$, since $\operatorname{tr}(A)=\sum_i\lambda_i=0$.

(2)

$\mathcal E(K_n)=2(n-1)$ for the complete graph on $n\geq2$ vertices.

(3)

$\mathcal E(K_{a,b})=2\sqrt{ab}$ for the complete bipartite graph with parts of sizes $a,b\geq1$.

(4)

$\mathcal E(P_n)=2\sum_{j=1}^{n}|\cos(j\pi/(n+1))|$ for the path on $n\geq2$ vertices.

(5)

$\mathcal E(C_n)=2\sum_{j=0}^{n-1}|\cos(2\pi j/n)|$ for the cycle on $n\geq3$ vertices.

(6)

$2\sqrt m\leq\mathcal E(G)\leq\sqrt{2mn}$, where $n$ is the number of vertices and $m$ is the number of edges. The lower equality holds for complete bipartite graphs [1]; the upper bound follows from $\sum_i\lambda_i^2=2m$ and the Cauchy-Schwarz inequality.

Comments

(7)

The adjacency matrix $A$ has $A_{uv}=1$ when $u$ and $v$ are adjacent and $A_{uv}=0$ otherwise. Its real eigenvalues are ordered $\lambda_1\geq\cdots\geq\lambda_n$, and $m=|E(G)|$ denotes the number of edges.

(8)

Each identifier is the graph6 encoding [2] of G.canonical_label(algorithm='sage'), computed with SageMath 10.9. Decode it with Graph(key, format='graph6'), preserving literal backslashes. These identifiers agree with the adjacency characteristic polynomial table, the Laplacian polynomial table, the spectral radius table and the Kirchhoff index table.

(9)

Entries are ordered by the number of vertices and then by ASCII order of the graph6 strings. Complete graphs, complete bipartite graphs, paths and cycles are named in entry comments.

(10)

The numbers of connected simple graph isomorphism classes at orders $2,3,4,5,6$ are $1,2,6,21,112$, respectively [3].

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.graphs.graph import Graph
from sage.rings.qqbar import AA
from sage.rings.real_mpfi import RealIntervalField

G = Graph('CL', format='graph6')  # the path on four vertices
I = RealIntervalField(numberdb.bits(100, losing=80))
spectrum = G.adjacency_matrix().charpoly().roots(ring=AA)
E = sum((multiplicity * abs(I(root)) for root, multiplicity in spectrum), I(0))
E

Links

[1]

Das and Gutman: Bounds for the energy of graphs, Sections 1 and 3

[2]

Brendan McKay: graph6 format specification

[3]

Brendan McKay: connected simple graph catalogues

Similar tables

Characteristic polynomials of connected graphs — Their roots, with multiplicity, are the eigenvalues used to compute graph energy.

Spectral radii of connected graphs — The largest adjacency eigenvalue is one of the terms in the positive-eigenvalue sum.

Laplacian polynomials of connected graphs — The same graph identities index the characteristic polynomials of the combinatorial Laplacian.

Kirchhoff indices of connected graphs — The same graph identities index the total effective resistance.

Estrada indices of connected graphs — Uses the same adjacency spectrum, summing its exponentials.

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds all $142$ isomorphism classes of connected simple graphs with $2\leq|V(G)|\leq6$. This exhaustive small-order census covers trees and cyclic graphs, bipartite and non-bipartite graphs, and regular and irregular graphs, including energies with several positive and negative spectral modes. Six is a practical, noncanonical boundary independent of numerical notation; order seven would add $853$ classes. No separate application is claimed for every graph)

How they were obtained:

The generator factors the integer adjacency characteristic polynomial over the rationals, isolates its real roots in Sage's AA, and determines their signs exactly. An irreducible factor whose roots have one sign contributes its signed integer trace with its multiplicity. These contributions give 13 exact integer energies. The remaining 129 energies are summed in RealIntervalField at numberdb.bits(100, losing=80), or 413 bits. Every interval has width less than $2.27\times10^{-123}$; no precision increase was needed. Each entire certified interval fits inside the corresponding written interval of 100 significant digits.

A rational total cannot be hidden among the mixed-sign cases. A Galois conjugate of the signed eigenvalue sum is at most the energy, term by term. Transitivity on a mixed-sign irreducible factor gives a conjugate with a strictly smaller value, ruling out rationality. Thus exactness is decided algebraically, without rounding the numerical sum to an integer.

An independent program decodes graph6 directly and diagonalizes the resulting integer matrices with mpmath at 150 and 180 decimal digits. All 142 values agree between those precisions to within $3.05\times10^{-150}$ and agree with the certified values. This is an independent numerical agreement check; the proof of the stored error bounds comes from the algebraic isolation and interval arithmetic. Controls use an edgeless graph, $K_2$ with energy $2$, $K_3$ with energy $4$, and the graph6 specification's decoding example.

Exact closed-walk traces and Newton identities reproduce every characteristic polynomial in the adjacency characteristic polynomial table. The graph census agrees as a set with McKay's catalogue files [3], and every key agrees with the order-two-through-six subsets of T237, T238, T397 and T400 and with T545. All five complete graphs, nine unordered complete-bipartite part-size pairs, five paths and four cycles satisfy their stated formulas; the trace identities, positive-eigenvalue formula and energy bounds hold throughout the census. The attached checks.py repeats the comparisons on the stored values, and generate.py verifies every entry. The attached argument-selection.md records the parameter justification and precision measurements.